Asymptotic expansion formulas for functionals of ε-Markov processes with a mixing property

The ε-Markov process is a general model of stochastic processes which includes nonlinear time series models, diffusion processes with jumps, and many point processes. With a view to applications to the higher-order statistical inference, we will consider a functional of the ε-Markov process admitting a stochastic expansion. Arbitrary order asymptotic expansion of the distribution will be presented under a strong mixing condition. Applying these results, the third order asymptotic expansion of theM-estimator for a general stochastic process will be derived. The Malliavin calculus plays an essential role in this article. We illustrate how to make the Malliavin operator in several concrete examples. We will also show that the thirdorder expansion formula (Sakamoto and Yoshida (1998, ISM Cooperative Research Report, No. 107, 53–60; 1999, unpublished)) of the maximum likelihood estimator for a diffusion process can be obtained as an example of our result.     

Random evolutions with locally independent increments on increasing time intervals

Three main schemes of limit theorems for random evolutions are discussed: averaging, diffusion approximation, and the asymptotics of large deviations. Markov stochastic evolutions with locally independent increments on increasing time intervals T _ε  = t/ε → ∞, ε → 0, are considered. The asymptotic behavior of random evolutions is investigated with the use of solutions of the singular perturbation problems for reducibly invertible operators.     

Refined asymptotics for constant scalar curvature metrics with isolated singularities

We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, , in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, [2], we present a much simpler and more geometric derivation of this fact. We also discuss a refinement, showing that any such solution is asymptotic to one of the deformed radial singular solutions. Finally we give some applications of these refined asymptotics, first to computing the global Pohožaev invariants of solutions on the sphere with isolated singularities, and then to the regularity of the moduli space of all such solutions. Oblatum 26-II-1997 & 6-II-1998 / Published online: 12 November 1998     

Malliavin calculus and martingale expansion

We proved the validity of the asymptotic expansion for the distribution of a martingale with jumps. A sufficient condition is presented in terms of the decay of certain integrations of Fourier type. In order to estimate such Fourier type integrals, we use the Malliavin calculus and show that it becomes a key to our program.     

The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point

We consider the Markov diffusion process ξ^∈(t), transforming when ɛ=0 into the solution of an ordinary differential equation with a turning point ℴ of the hyperbolic type. The asymptotic behevior as ɛ→0 of the exit time, of its expectation of the probability distribution of exit points for the process ξ^∈(t) is studied. These indicate also the asymptotic behavior of solutions of the corresponding singularly perturbed elliptic boundary value problems.     

Asymptotic behavior of positive solutions to emden–fowler type equations

We study the asymptotic behavior of positive solutions to nonlinear elliptic equations of Emden–Fowler type with absorption term. For operators with variable coefficients we obtain conditions on coefficients under which the solutions have the same asymptotics as solutions to the model equation Δu = −x| ^p |u| ^σ−1 u. For positive solutions we obtain lower order terms of the asymptotic expansion at infinity. Bibliography: 10 titles.     

Third-order asymptotic expansion of <Emphasis Type="Italic">M</Emphasis>-estimators for diffusion processes

For an unknown parameter in the drift function of a diffusion process, we consider an M-estimator based on continuously observed data, and obtain its distributional asymptotic expansion up to the third order. Our setting covers the misspecified cases. To represent the coefficients in the asymptotic expansion, we derive some formulas for asymptotic cumulants of stochastic integrals, which are widely applicable to many other problems. Furthermore, asymptotic properties of cumulants of mixing processes will be also studied in a general setting.     

Local limit theorems for transition densities of Markov chains converging to diffusions

We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Local limit theorems for transition densities are proved. Received: 28 August 1998 / Revised version: 6 September 1999 / Published online: 14 June 2000     

Importance and Sensitivity Analysis in Dynamic Reliability

In dynamic reliability, the evolution of a system is governed by a piecewise deterministic Markov process, which is characterized by different input data. Assuming such data to depend on some parameter p ∈ P, our aim is to compute the first-order derivative with respect to each p ∈ P of some functionals of the process, which may help to rank input data according to their relative importance, in view of sensitivity analysis. The functionals of interest are expected values of some function of the process, cumulated on some finite time interval [0,t], and their asymptotic values per unit time. Typical quantities of interest hence are cumulated (production) availability, or mean number of failures on some finite time interval and similar asymptotic quantities. The computation of the first-order derivative with respect to p ∈ P is made through a probabilistic counterpart of the adjoint state method, from the numerical analysis field. Examples are provided, showing the good efficiency of this method, especially in case of a large P.     

On a Unique Ergodicity of Some Markov Processes

It is proved that the sufficient condition for the uniqueness of an invariant measure for Markov processes with the strong asymptotic Feller property formulated by Hairer and Mattingly (Ann Math 164(3):993–1032, 2006) entails the existence of at most one invariant measure for e-processes as well. Some application to time-homogeneous Markov processes associated with a nonlinear heat equation driven by an impulsive noise is also given.     

Diffusion processes on graphs: stochastic differential equations,large deviation principle

Ito's rule is established for the diffusion processes on the graphs. We also consider a family of diffusions processes with small noise on a graph. Large deviation principle is proved for these diffusion processes and their local times at the vertices. Received: 12 February 1997 / Revised version: 3 March 1999     

Asymptotic analysis of the steady stokes equation with randomly perturbed viscosity in a thin tube structure

The Stokes equation with the nonconstant viscosity is considered in a thin tube structure, i.e., in a connected union of thin rectangles with heights of order ε ≪ 1 and bases of order 1 with smoothened boundary. An asymptotic expansion of the solution is constructed. In the case of random perturbations of the constant viscosity, we prove that the leading term for the velocity is deterministic, while for the pressure it is random, but the expectations of the pressure satisfies the deterministic Darcy equation. Estimates for the difference between the exact solution and its asymptotic approximation are proved. Bibliography: 11 titles. Illustrations: 3 figures.     

Martin–Kuramochi boundary and reflecting symmetric diffusion

In this paper, we will give sufficient conditions for the existence of the reflecting diffusion process on a locally compact space. In constructing reflecting diffusion process, we consider the corresponding Martin–Kuramochi boundary as the reflecting barrier and introduce the notion of strong (ℰ, u)-Caccioppoli set. Our method covers reflecting diffusion processes with diffusion coefficient degenerating on the boundary. Received: 23 June 1997 / Revised version: 28 September 1991/ Published online: 14 June 2000     

Extrapolation algorithms for solving nonlinear boundary integral equations by mechanical quadrature methods

We study the numerical solution procedure for two-dimensional Laplace’s equation subjecting to non-linear boundary conditions. Based on the potential theory, the problem can be converted into a nonlinear boundary integral equations. Mechanical quadrature methods are presented for solving the equations, which possess high accuracy order O(h ³) and low computing complexities. Moreover, the algorithms of the mechanical quadrature methods are simple without any integration computation. Harnessing the asymptotical compact theory and Stepleman theorem, an asymptotic expansion of the errors with odd powers is shown. Based on the asymptotic expansion, the h ³ −Richardson extrapolation algorithms are used and the accuracy order is improved to O(h ⁵). The efficiency of the algorithms is illustrated by numerical examples.     

Intertwining Certain Fractional Derivatives

We obtain an intertwining relation between some Riemann–Liouville operators of order α ∈ (1, 2), connecting through a certain multiplicative identity in law the one-dimensional marginals of reflected completely asymmetric α-stable Lévy processes. An alternative approach based on recurrent extensions of positive self-similar Markov processes and exponential functionals of Lévy processes is also discussed.     

Milnor K-theory of rings,higher Chow groups and applications

If R is a smooth semi-local algebra of geometric type over an infinite field, we prove that the Milnor K-group K ^M _n (R) surjects onto the higher Chow group CH ⁿ (R , n) for all n≥0. Our proof shows moreover that there is an algorithmic way to represent any admissible cycle in CH ⁿ (R , n) modulo equivalence as a linear combination of “symbolic elements” defined as graphs of units in R. As a byproduct we get a new and entirely geometric proof of results of Gabber, Kato and Rost, related to the Gersten resolution for the Milnor K-sheaf. Furthermore it is also shown that in the semi-local PID case we have, under some mild assumptions, an isomorphism. Some applications are also given. Oblatum 17-XII-1998 & 1-X-2001?Published online: 18 January 2002     

On logarithmic Sobolev inequalities for continuous time random walks on graphs

We establish modified logarithmic Sobolev inequalities for the path distributions of some continuous time random walks on graphs, including the simple examples of the discrete cube and the lattice ZZ ^d . Our approach is based on the Malliavin calculus on Poisson spaces developed by J. Picard and stochastic calculus. The inequalities we prove are well adapted to describe the tail behaviour of various functionals such as the graph distance in this setting. Received: 6 April 1998 / Revised version: 15 March 1999 / Published on line: 14 February 2000